1. STAT 250 Dr. Kari Lock Morgan
Multiple Regression
SECTIONS 10.1, 10.3
Multiple explanatory variables (10.1, 10.3)

2. More than 2 variables!
For the rest of the course, we’ll finally get beyond one or two variables!

3. How can we predict body fat percentage from easy measurements?
Question of the Day
How can we predict body fat percentage from easy measurements?
pheromones = subconscious chemical signals

4. Predicting Body Fat Percentage
The percentage of a person’s weight that is made up of body fat is often used as an indicator of health and fitness
Accurate measures of percent body fat are hard to get (for example, immerse the body in water to estimate density, then apply a formula)
Another option: build a model predicting % body fat based on easy to obtain measurements

5. Body Fat Data Measurements were collected on 100 men
Response variable: percent body fat
Explanatory variables:
Age (in years)
Weight (in pounds)
Height (in inches)
Neck circumference (in cm)
Chest circumference (in cm)
Abdomen circumference (in cm)
Ankle circumference (in cm)
Biceps circumference (in cm)
Wrist circumference (in cm)
A sample taken from data provided by Johnson R., "Fitting Percentage of Body Fat to Simple Body Measurements," Journal of Statistics Education, 1996

6. Multiple Regression
Multiple regression extends simple linear regression to include multiple explanatory variables:
Each x is a different explanatory variable
k is the number of explanatory variables

7. Three Explanatory Variables
We’ll start with three explanatory variables: age, weight, height The regression equation is…
𝐵𝑜𝑑𝑦𝑓𝑎𝑡 = 𝐴𝑔𝑒+0.023𝑊𝑒𝑖𝑔ℎ𝑡+.256𝐻𝑒𝑖𝑔ℎ𝑡
𝐵𝑜𝑑𝑦𝑓𝑎𝑡 =0.05𝐴𝑔𝑒+0.023𝑊𝑒𝑖𝑔ℎ𝑡+.256𝐻𝑒𝑖𝑔ℎ𝑡
𝐵𝑜𝑑𝑦𝑓𝑎𝑡 = 𝐴𝑔𝑒 𝑊𝑒𝑖𝑔ℎ𝑡−1.1169𝐻𝑒𝑖𝑔ℎ𝑡
𝐵𝑜𝑑𝑦𝑓𝑎𝑡 =0.1653𝐴𝑔𝑒 𝑊𝑒𝑖𝑔ℎ𝑡−1.1169𝐻𝑒𝑖𝑔ℎ𝑡

8. Predicting Percent Body Fat
What can we do with this?
Make predictions
Interpret coefficients
Inference
Interpret R2
and more!

9. Making Predictions
If you are male, you can use this to predict your percent body fat!
(Females can try too, just for practice, but it won’t be accurate – why not?)
Age: years, weight: pounds, height: inches

10. Percent Body Fat

11. Interpreting Coefficients
Intercept: a man 0 years old, weighs 0 lbs, and is 0 inches tall would have 49.6% body fat
Slope: Keeping weight and height constant, percent body fat increases by for every additional year
Keeping age and height constant, percent body fat increases by for every additional pound

12. Interpreting Coefficients
Which of the following is a correct interpretation?
Keeping age and weight constant, height decreases by for every additional percent of body fat
Keeping age and weight constant, percent body fat decreases by for every additional inch
Predicted body fat decreases by for every additional inch

13. Minitab Output

14. Inference Are our explanatory variables significant predictors?
All of the p-values corresponding to the explanatory variables are very small
Age, weight, and height are all significant predictors of percent body fat (given the other variables in the model)

15. R2
R2 is the proportion of the variability in the response variable, Y, that is explained by the fitted model
For simple linear regression, R2 = r2 (R2 is just the sample correlation squared)
R2 is also called the coefficient of determination

16. R2
How much does the variability in Y decrease if you know X?

17. R2
About 55% of the variability in percent body fat is explained by age, weight, and height
Can we do better?

18. Comparing with BMI
BMI is used more commonly than percent body fat because it is easy to calculate
Currently, our predicted percent body fat is not using much more information than BMI (just age as an extra predictor)
What’s wrong with body mass index (BMI) as a indicator of health and fitness?
How might we improve our model to fix this problem?

19. New Model
Bodyfat =  0.0067 Age - 0.1724 Weight + 0.099 Height + 1.066 Abdomen
Anything look odd about this equation???
Model without Abdomen: Bodyfat =  0.1653 Age + 0.2264 Weight - 1.117 Height
What’s going on?!?

20. Significance Which explanatory variable(s) are significant?
All of them – age, weight, height, abdomen
Weight and height
Weight, height, abdomen
Weight and abdomen
Abdomen only

21. Multiple Regression
The coefficient for each explanatory variable is the predicted change in y for one unit change in x, given the other explanatory variables in the model!
The p-value for each coefficient indicates whether it is a significant predictor of y, given the other explanatory variables in the model!
If explanatory variables are associated with each other, coefficients and p-values will change depending on what else is included in the model

22. Full Model

23. Which explanatory variable(s) are significant?
All of them
Weight and abdomen
Neck only
Abdomen and wrist

24. Insignificant Terms
What should we do with the insignificant variables?
Keep them in the model?
Take them out of the model?
Deciding which variables to keep in the model (variable selection) is an entire subfield of statistics, and beyond the scope of this class
Want to learn more about it? Take STAT 462!

25. Explaining Variability
How much of the variability in percent body fat is explained by this model? Which of the following would tell us this?
p-value
correlation
slope coefficients R2
confidence interval

26. Full Model

27. What will I get on the final exam???
Question #2 of the Day
What will I get on the final exam???
pheromones = subconscious chemical signals

28. Model Output All grades are in percent form (0 – 100)
You can predict your final exam score based on your performance so far!
You have a point estimate… what do you really want???

29. Uncertainty?
To get an exact prediction interval, use “Predict for Regression” in Minitab
To get an approximate interval, take your predicted value and add and subtract 2 × 𝑆

30. Significance
WileyPlus and Clicker grades are not significant in the model. Does this mean that they are not significantly associated with Final Exam score?
Yes No

31. Significance
Clicker is still not significant in the model. Does this mean coming to class doesn’t matter?
Yes No

32. Clicker
Can we conclude that coming to class improves your score on the final exam?
Yes No

33. Multiple Regression
Coefficients and p-values depend on the other explanatory variables included in the model!!!

34. Lots More!
The goal of this class was to expose you to multiple regression as a way to incorporate more than two variables
This one class does NOT cover everything you should know about regression!
If you really want to use multiple regression for data analysis, take STAT 462! (Or consult with a statistician)

35. To Do
Do HW (due Wednesday, 4/19)